How do I stop the UV map from changing while adding bevels?
The reason this is happening is because you are beveling an edge with a seam on it. Since beveling creates multiple edges from a single edge, blender has to place the seam somewhere, yet there is no 'right' edge to place the seam on, so it gets applied to both created edges (extra segments do not receive a seam). What that means is that you will have to do some manual adjustments to make this work how you want. I would suggest removing the seam first, then beveling, then re-adding the seam where you want it. An alternative is to remove the seams after you have beveled the edges, but that will probably be a bit harder to do.